Integrating the topics of forms, manifolds, and algebra

[observer1]

Hello,

As you might discern from previous posts, I have been teaching myself:

1. Calculus on manifolds
2. Differential forms
3. Lie Algebra, Group
4. Push forward, pull back.

I am an engineer approaching this late in life and with a deficient background in math. It is all coming together and I almost see the picture. But it remains discrete topics - not connected with each other and without a direct connection to my deficienies.

This post concerns THE GLUE.

Could someone explain how all these ideas are integrated, in WORDS and with an acknowledgment to a person with deficient math (math learned in a country with a math education a mile wide and an inch deep).

Essentially, IN WORDS (not symbols)...
And YES, these are huge questions, but I am not interested in direct answers to each but in the continuity between the questions -- oh hell, I wish I knew exactly how to say this: my calculus was so bad I practically need someone to say: "In a Lagrangian there are generalized coordinates which become the manifold and there are vectors in the tangent plane (position) and can only be multiplied by co-vectors (velocities) and the purpose of all this is only to see the structure..." Or something like that...

Thus...

Why should I learn calculus on manifolds (ConM)?

What impact do they have in Lagrange's equation for a dynamical system?

What can ConM do that my “normal” calculus cannot?

Why do forms matter?

How do form really help do math on the tangent space?

What is the relationship between forms and vectors?

Why is it that all treatments on this also treat Lie Algebra and Groups?

Lie groups are just smooth groups, so where is the manifold.?I know it is going to be difficult to do this – write an answer in words.

Thank you if you can. (And, yes, as evidenced by previous posts: I am looking for words... for only by WORDS can I see where I was miseducated.)

 

[fresh_42]

Hello,

As you might discern from previous posts, I have been teaching myself:

1. Calculus on manifolds
2. Differential forms
3. Lie Algebra, Group
4. Push forward, pull back.

I am an engineer approaching this late in life and with a deficient background in math. It is all coming together and I almost see the picture. But it remains discrete topics - not connected with each other and without a direct connection to my deficienies.

This post concerns THE GLUE.

Could someone explain how all these ideas are integrated, in WORDS and with an acknowledgment to a person with deficient math (math learned in a country with a math education a mile wide and an inch deep).

Essentially, IN WORDS (not symbols)...
And YES, these are huge questions, but I am not interested in direct answers to each but in the continuity between the questions -- oh hell, I wish I knew exactly how to say this: my calculus was so bad I practically need someone to say: "In a Lagrangian there are generalized coordinates which become the manifold and there are vectors in the tangent plane (position) and can only be multiplied by co-vectors (velocities) and the purpose of all this is only to see the structure..." Or something like that...

Thus...

Why should I learn calculus on manifolds (ConM)?

What impact do they have in Lagrange's equation for a dynamical system?

What can ConM do that my “normal” calculus cannot?

Why do forms matter?

How do form really help do math on the tangent space?

What is the relationship between forms and vectors?

Why is it that all treatments on this also treat Lie Algebra and Groups?

Lie groups are just smooth groups, so where is the manifold.?I know it is going to be difficult to do this – write an answer in words.

Thank you if you can. (And, yes, as evidenced by previous posts: I am looking for words... for only by WORDS can I see where I was miseducated.)

Your questions are difficult if not impossible to answer without writing a book!
I give it a try to answer them as short as they are.

Why should I learn calculus on manifolds (ConM)?

This depends on what you want to learn at all and what you are planning to do with it.
Modern physics depend on it. It basically started with the attempt to solve differential equations which describe a whole lot of nature's behavior: from the outbreak of pandemics over the population of species to the fabric of spacetime .

What impact do they have in Lagrange's equation for a dynamical system?

(see above; beside that I recommend a physicist or biologist to answer this)

What can ConM do that my “normal” calculus cannot?

To me it is normal calculus. It may help to solve (systems of) differential equations. The prediction of recently found gravitational waves is an example. The surface of our Earth is a manifold. Even your "normal" real vector world can be considered as a manifold. I guess people started to think: why restrict to a flat world if a) the world isn't and b) results can simultaneously be proven in a more general case. It is like the question: why should I calculate the volume of a sphere if I already know the volume of a cube?

Why do forms matter?

It's a generalization of derivatives.

How do form really help do math on the tangent space?

It is math on/with the tangent space.

What is the relationship between forms and vectors?

What is the relationship between functions and values?

Why is it that all treatments on this also treat Lie Algebra and Groups?

Lie groups are manifolds, Lie algebras are tangent spaces. E.g. they arise from the examination of symmetries, rotations and so on.
In particle physics you can't do a step without meeting the small ones of them. They are also simply useful examples: rotations in a plane are imaginable, a string world in eleven dimensions not really.

Lie groups are just smooth groups, so where is the manifold.?

Right in front of you. A path through elements of a Lie group is a path on a manifold. It is a curvy thing.

I apologize for being sloppy on some points. As I said: you can fill a book and actually there are dozens of books dealing with this stuff. However, I admit most scientist are weak when it comes to explain why and what for. Usually they put it in exercises or simple examples and hope the results themselves explain the necessity. Beside that mathematics from a physical point of view is a big toolbox and you never know in advance which tool you need so the more the better is the motto.

 

[observer1]

OK then.. so I know the gradient operator has different forms in different coordinate systems. I get that.

So when someone does Calculus on Manifolds (say the configuration space of a pendulum in which it is reduce not to x/y position in Cartesian,but to THETA, say in generalized), does it all come down to recognizing that derivatives are DIFFERENT in these manifold subspaces? Is that all it is?

See, I sort of get the idea of manifolds, but I don't get why people make a big deal on doing calculus on them when I have no issue about the gradient in other coordinate systems. Or are you saying that THE REASON I have no issue is that I have intrinsically ACCEPTED the idea of calculus on manifolds?

When you say: “What is the relationship between functions and values?

Could you expand with a statement and not a question?
Are you saying form is to vector as functions are to their values?
I am confused.

I just need someone to say: "when we say X, we mean it in relation to THE THING you learned poorly in Calculus in Euclidean space.

 

[fresh_42]

OK then.. so I know the gradient operator has different forms in different coordinate systems. I get that.

So when someone does Calculus on Manifolds (say the configuration space of a pendulum in which it is reduce not to x/y position in Cartesian,but to THETA, say in generalized), does it all come down to recognizing that derivatives are DIFFERENT in these manifold subspaces? Is that all it is?

See, I sort of get the idea of manifolds, but I don't get why people make a big deal on doing calculus on them when I have no issue about the gradient in other coordinate systems. Or are you saying that THE REASON I have no issue is that I have intrinsically ACCEPTED the idea of calculus on manifolds?

When you say: “What is the relationship between functions and values?

Could you expand with a statement and not a question?
Are you saying form is to vector as functions are to their values?
I am confused.

I just need someone to say: "when we say X, we mean it in relation to THE THING you learned poorly in Calculus in Euclidean space.

No, now I'm a bit confused. Maybe I didn't understand what you've meant by 'form'. I was referring to a differential form which is a function on vector(field)s generalizing (directional) derivations.

Concerning the coordinate system one has to distinguish between the object itself and its representation. A pendulum doesn't change whether you describe it in cartesian or polar coordinates. It's the equations that look different. "I'm hungry" takes a different form if I say it in my own language. However, it doesn't change my status if I do so.

Whether manifolds are a big deal or not depends mainly on your attitude towards them. Do you remember your schooldays were negative numbers became from impossible to natural just by changing the grade? Manifolds stand just for something curvy (even if the curvature vanishes). Curves are everywhere in this world. We consider tangent spaces as a local approximation of these curves because they are easier to handle and locally as good as the curve itself. It is the curve sketching from school in higher dimensions and on different surfaces, e.g. analytic groups.

 

[mathwonk]

Riemann’s idea was to represent physics through geometry. The geometry of space is essentially a curved manifold, hence we study those objects, e.g. a sphere. To study an object one looks for its intrinsic nature and properties. A smooth geometric space is one which is locally approximable by flat spaces, i.e. its tangent spaces at each point, so it behooves us to study the tangent spaces to our manifold. This is also where the velocity vectors to curves live, hence of importance to physicists. Analyzing those tangent spaces involves understanding families of their elements, i.e. vector fields. This is the sort of thing introduced into space by forces like gravity.

Studying those involves also understanding functions on them, and the linear functions on vectors are called “forms”, and families or fields of those linear functions are called differential forms. Differential forms can be integrated and allow us thereby to measure physical and geometric quantities like angles, as when integrating the form dtheta. Functions having values on our manifold also inform us about physical quantities like temperature, and each function has a differential, which is an “exact” differential form and whose variation can be measured by its integrals along curves.

Your ordinary calculus is the basis for all this, but is usually restricted to flat or euclidean spaces, whereas the theory of manifolds expands its scope to curved ones. So calculus on manifolds expands and generalizes the methods of ordinary calculus to apply to curved spaces, preserving the intrinsic properties of derivatives and integrals, allowing us to discuss them with less dependence on specific coordinates.

The more intrinsic structure of an object we discover the more we can study it. Viewing tangent vectors via their action as differential operators, i.e. taking directional derivatives, allows us to define a sort of multiplication on vector fields, the lie bracket. We alaso enhance our knowledge of any space by studying itself transformations, and the transformations can be composed yielding a group structure. E.g. the group of matrices, or linear transformations of euclidean space, is also a smooth mnifold and admits study by calculus. morever any group allows us to translate all vectors to the origin, and hence replace vector fields by just vectors at the origin. thus the lie operation on vector fields allows us to define a lie group structure on the tangent space at the origin of a smooth manifold which is a group. so manifolds lead to lie groups and to lie algebras.

forgive my errors, they are due to ignorance; this is a big question but i tried to say something of use.

 

[The Bill]

What can ConM do that my “normal” calculus cannot?

The calculus of differential forms let's you do "vector calculus" integrals on spaces that don't let you use the cross product, like spaces with four or more dimensions. Also, knowing the generalized Stokes' theorem plus the rules of manipulating differential forms means you never have to look up any of the integral theorems from traditional "vector calculus" again, because you can just look at the problem, write out the integral, and immediately plug in the values and start calculating it or deriving its properties.

What impact do they have in Lagrange's equation for a dynamical system?

The phase space for the Hamiltonian formulation of a system in classical mechanics with a finite number of degrees of freedom can be viewed as a symplectic manifold. The techniques of differential geometry, including what is classed as calculus on manifolds, can then be used to examine the invariants and phase portraits of the system. V.I. Arnold's Mathematical Methods of Classical Mechanics is devoted to teaching this framework for classical mechanics.

The book that taught me the "glue" between the topics you mentioned in the first post is William L. Burke's Applied Differential Geometry.

 

[Twigg]

Why should I learn calculus on manifolds (ConM)?

What impact do they have in Lagrange's equation for a dynamical system?

What can ConM do that my “normal” calculus cannot?

I've been studying differential geometry for 3-4 years now, and I can count on one hand the number of times it's been essential in solving a practical problem in physics or engineering. The examples that come to mind are the Foucault pendulum (a basic grasp of Lie theory allows you to transform to the rotating frame at the initial point), calculating the precession of mercury (the metric tensor showed up in the first few lines, that's it), and making sense of time evolution operators and operator algebra in quantum mechanics (another application of basic Lie theory). There are far more problems that fully-general differential geometry can be applied to than it must be applied to. My advice for problem solving is to avoid the general mathematical theories of differential geometry like the plague wherever possible. The tell-tale sign of a problem that necessitates the use of methods beyond the scope of calculus on Euclidean space is the absence of any global coordinate chart. These problems are truly rare and extremely messy. In physics, I've only encountered them in problems involving the use of non-inertial or gravitational frames or an interaction picture. In applied mathematical problems, I've also run into them in the analysis of symmetries of differential and algebraic equations and variational/optimization problems when you look at solution in a neighborhood of a point without any prior knowledge of the shape of the hypersurface that solves your problem. For example, suppose I wanted to know the radiation field produced by a thin rotating dipole antenna. I can use a Lie group created by the rotating frame of the antenna to connect a surface element of a wavefront to an inertial frame, in which I know exactly how the radiation field will evolve using Green's functions.

Noether's theorem in Lagrangian mechanics is based on Lie theory. In theory, all of variational calculus is applied Lie theory, but that doesn't make it any more powerful than it was when the Bernoullis, Euler, Fermat, Lagrange, Hamilton, and others developed the subject well before Lie's publications. I single out Noether's theorem in particular because it involves the direct use of symmetry generators to demonstrate.

The phase space for the Hamiltonian formulation of a system in classical mechanics with a finite number of degrees of freedom can be viewed as a symplectic manifold. The techniques of differential geometry, including what is classed as calculus on manifolds, can then be used to examine the invariants and phase portraits of the system.

I would point out that the whole point of Hamiltonian mechanics is to provide you with a Poisson bracket so you don't have to deal with the full differential geometric analysis of invariants over jet bundles. For instance, if you want to find invariants in a system in Lagrangian mechanics, you have to take a first-order partial differential operator on the tangent bundle of position space (not a vector field in the tangent bundle, a vector field ON the whole tangent bundle). This requires some algebraic heavy lifting, as their are intricate expressions constraining differential operators on the tangent bundle obtained by prolonging transformations on position space. The Poisson bracket is a convenient short-cut to the analysis of invariants that really depends on nothing but the chain rule on phase space and Hamilton's equations. The development of the fully general Poisson bracket is demonstrated without reference to differential geometry (except for a passing comment in retrospect about local contact transformations at the end) in the second section of Dirac's Lectures on Quantum Mechanics.

Why do forms matter?

How do form really help do math on the tangent space?

As stated above, the general theory of differential forms isn't necessary until you start talking about a hypersurface that doesn't have any global coordinate charts, which forces you to state your problem locally (i.e., your problem becomes a family of subproblems each restricted to the domain of a certain local coordinate chart). In the problem I stated above about the thin rotating dipole antenna, the "subproblems" are the radiation field patterns around each surface element of the wavefront in question. You know the shape of the wavefronts of the radiation pattern in the inertial frame from integrating the Green's functions over the length of the antenna. Surface elements of the wavefront (which themselves are very simplistic differential 2-forms) can be "pulled back" to more complicated locally defined differential forms in the rotating frame via the inverse of the rotation transformation that took us to the inertial frame in the first place. That is a case where the full theory of differential forms would mean something more complicated than you'd see in ordinary calculus on Euclidean space, because you're working in more than three dimensions on a geometric shape you pretend to know nothing about except that it's locally related to the inertial frame by a transformation that varies from point to point.

Forms do math on the tangent space in the same way the a surface element acts on an electric field in Gauss's law. They enable you to evaluate fluxes through hypersurfaces by integrating vector fields and higher dimensional "geometric flux densities." In fancy lingo, these flux densities are contravariant tensor densities, such as the electromagnetic stress-energy tensor in relativistic electrodynamics ($T^{\mu\nu}$), which is like a 4-dimensional generalization of the Maxwell stress tensor in 3-dimensional electrodynamics, or the Faraday electromagnetic field tensor ($F^{\mu\nu}$), which is like the 4-dimensional analogue of the electric flux density ($\vec{D}$) and magnetic flux density ($\vec{B}$) combined. If the basis differential forms are n-1 dimensional surface elements and parallelotopes formed with parallel families of surface elements for edges ($dx \wedge dy$ is like taking the flux of a field through level sets of x followed by taking the flux through level sets of y on the boundary of an infinitesimal rectangle with edge of length |dx| and |dy|), then the action of these "generalized oriented differential parallelotopes" on a same-dimensional flux density is the flux of said field around the parallelotope.

What is the relationship between forms and vectors?

If vector fields represent slope fields on your space, then 1-forms represent a field of infinitesimal flat hyperplanes that appear locally parallel to each other and everywhere normal to the vector field to which they are dual. If you happen to have a metric tensor (which is guaranteed on smooth manifolds of finite dimension), then duality happens to coincide with index raising/lowering operations because of the wonders of doing linear algebra in an inner product space. Tensorial acrobatics aside, the relationship between forms and vectors is that of orthogonal complement (which as linear algebra operation in flat space would be expressed as the transpose). This is why surface integrals in 3D Euclidean space can be represented as a magnitude times a normal vector.

Why is it that all treatments on this also treat Lie Algebra and Groups?

Lie groups are just smooth groups, so where is the manifold.?

Lie algebras and Lie groups enjoy a prominent role in differential geometric theory because on any space, curved or flat, (of finite dimension, as far as I know), every vector field is an infinitesimal transformation. More formally, the set of all vector fields in the tangent bundle of any smooth manifold (of finite dimension) belongs to an infinite dimensional Lie algebra. The corresponding one-parameter Lie group associated with a given vector field is the group of translations along integral curves of that field. It turns out you can't do or prove much of anything without invoking this particular Lie algebra. The basic reason for this is that this Lie group, of translations along curves, is what connects geometric points to their neighboring points without leaving the manifold; it represents motion restricted to the manifold, at least locally. How would you even begin to talk about tangent spaces, let alone metrics, if you couldn't connect points? (If you have a local transformation group connecting points in a neighborhood, then the differential, or "push-forward" of that transformation connects tangent planes of these points).

Smoothness doesn't mean anything if it isn't on a manifold. Globally-defined Lie groups (as opposed to local Lie groups, which are much less common in modern treatments but what you're most likely to come across in a problem that needs Lie theory to solve) are usually written in terms of parameters. For example, SO(2) is written in terms of the parameter $\theta$ which identifies each element of SO(2) with an element of the unit circle and gives SO(2) the same smooth structure as the unit circle. SO(3) has the same smooth structure as the real projective space $\mathbb{RP}^{3}$. SU(2) has the same structure as the 3-sphere. In other words, if I pick a point on the 3-sphere (any point), I also pick a transformation in SU(2). Any calculus-ish statement about one can be taken as an identical statement about the other. The operation of group multiplication, the differentials of multiplication by elements, the Lie algebra, and anything else about the structure of SU(2) can be taken to mean something geometric on the 3-sphere.
I apologize for the lateness of the post. I put this up anyways because I felt this could be helpful to the OP and many others studying the field.

 

[lavinia]

Ordinary calculus in Euclidean space is calculus on manifolds, this because Euclidean space is a manifold. Euclidean space has different coordinate charts and coordinate transformations just as other manifolds.

Classical vector calculus uses the Euclidean metric in 3 dimensional space to translate differential forms into vectors. Without this metric (or some other metric), there would be no gradients or curls. There would only be 1 forms and 2 forms. The moral is: Calculus does not require a metric and a big part of learning calculus on manifolds is just to relearn ordinary vector calculus without using a dot product.

Perhaps the key idea that separates classical vector calculus from calculus on manifolds is not the calculus, but rather the manifold - as Mathwonk has explained above. In general, manifolds cannot be covered by a single coordinate chart as can Euclidean space and this is the big difference. Because of this vector fields and differential forms and tensors may not be describable in terms of a global basis the way they can in Euclidean space. This allows for many physical and mathematical phenomena that do not occur in a single coordinate chart. That by itself suffices as a reason to learn how to do calculus on them.

 

[observer1]

Riemann’s idea was to represent physics through geometry. The geometry of space is essentially a curved manifold, hence we study those objects, e.g. a sphere. To study an object one looks for its intrinsic nature and properties. A smooth geometric space is one which is locally approximable by flat spaces, i.e. its tangent spaces at each point, so it behooves us to study the tangent spaces to our manifold. This is also where the velocity vectors to curves live, hence of importance to physicists. Analyzing those tangent spaces involves understanding families of their elements, i.e. vector fields. This is the sort of thing introduced into space by forces like gravity.

Studying those involves also understanding functions on them, and the linear functions on vectors are called “forms”, and families or fields of those linear functions are called differential forms. Differential forms can be integrated and allow us thereby to measure physical and geometric quantities like angles, as when integrating the form dtheta. Functions having values on our manifold also inform us about physical quantities like temperature, and each function has a differential, which is an “exact” differential form and whose variation can be measured by its integrals along curves.

Your ordinary calculus is the basis for all this, but is usually restricted to flat or euclidean spaces, whereas the theory of manifolds expands its scope to curved ones. So calculus on manifolds expands and generalizes the methods of ordinary calculus to apply to curved spaces, preserving the intrinsic properties of derivatives and integrals, allowing us to discuss them with less dependence on specific coordinates.

The more intrinsic structure of an object we discover the more we can study it. Viewing tangent vectors via their action as differential operators, i.e. taking directional derivatives, allows us to define a sort of multiplication on vector fields, the lie bracket. We alaso enhance our knowledge of any space by studying itself transformations, and the transformations can be composed yielding a group structure. E.g. the group of matrices, or linear transformations of euclidean space, is also a smooth mnifold and admits study by calculus. morever any group allows us to translate all vectors to the origin, and hence replace vector fields by just vectors at the origin. thus the lie operation on vector fields allows us to define a lie group structure on the tangent space at the origin of a smooth manifold which is a group. so manifolds lead to lie groups and to lie algebras.

forgive my errors, they are due to ignorance; this is a big question but i tried to say something of use.

Thank you! A lot!
It is your simpler (not an insult here) that is what I was looking for...

"So calculus on manifolds expands and generalizes the methods of ordinary calculus to apply to curved spaces, preserving the intrinsic properties"

and

"Viewing tangent vectors via their action as differential operators, i.e. taking directional derivatives, allows us to define a sort of multiplication on vector fields, the lie bracket."

Those words helped me!

 

[lavinia]

"So calculus on manifolds expands and generalizes the methods of ordinary calculus to apply to curved spaces, preserving the intrinsic properties"

Calculus on manifolds does allow one to study curved spaces from an intrinsic viewpoint - that is, from inside the manifold without referring to an ambient Euclidean space. But a space is curved when it has a metric. A metric is an additional structure that is not needed in order to just do calculus on the manifold. One can differentiate and integrate without a metric.

There is an entire subject called Differential Topology that studies manifolds using only calculus. When one studies a curved space i.e. a manifold with a metric, the subject is called Differential Geometry. However the same differentiable manifold i.e. a manifold on which one can do calculus, can have infinitely many different metrics each giving a different curved shape to the same differentiable manifold.